# Analysis

There have been 9 completed talks and 7 topic suggestions tagged with analysis.

## Completed Talks

### Constructive analysis

Delivered by Fengyang Wang on Wednesday November 1, 2017

Constructive mathematics, as the name would suggest, is centered on the philosophy that mathematical proofs should be able to be turned into algorithms. We will contextualize constructive approaches to analysis, roughly following Bridges and Vîţă. This talk has no formal prerequisites beyond an elementary understanding of the real numbers and the usual concept of completeness. In particular, no logical background is assumed; intuitionistic logic will be overviewed in the talk. We will finish with a discussion of the ramifications of completeness of the real numbers.

A summary of this talk is available here.

### Cantor Set and Dynamical Systems

Delivered by James Bai on Friday March 31, 2017

The talk will be begin on the cosnstruction of the most commonly used tenary Cantor set. The talk will then talk about the common properties of Cantor set and methods of evaluating the size of the set. Then, depending on time, a brief introduction will be given on the dynamical system and chaos.

A summary of this talk is available here.

### $p$-adic Numbers

Delivered by Akshay Tiwary on Friday February 3, 2017

In this talk we will discover an alternate way to define the "size" of a ration al number. We will define the p-adic absolute value and see that this absolute value is Non-archimedean and that this fact leads to a a geometry that is very different from the geometry of the Real numbers (every p-adic triangle is isosc eles!). In addition I will show you some fun ideas from p-adic numbers (which I will denote by $\mathbb{Q}_p$ like the fact that $\sum_{n = 1}^{\infty} 3^n$ converges and how every element of $\mathbb{Q}_p$ has a base $p$ expansion. This is meant to be a very leisurely talk with almost no prerequisites and it should be fun so please come for it!

### Integrability of Riccati Equations

Delivered by Letian Chen on Friday January 27, 2017

In 1841, 3 years before Joseph Liouville discovered transcendental numbers, Joseph Liouville showed that the Riccati equation $y' = ay^2 + bx^m$ has a quadrature solution if and only if $m = 0, -2, -\frac{4n}{2n±1}$. Nowadays, we see Liouville's results as the foundation to qualitative analysis to differential equations, a fascinating subarea of DEs, which studies for example the existence and uniqueness of different kinds of equations. In my talk, I will (hopefully) prove the classical result of Liouville after a quick review of history and related linear theory.

### How to Complicate Fourier Analysis

Delivered by Mohamed El Mandouh on Friday December 2, 2016

Fourier analysis was initially introduced as a way to study the thermodynamic heat equation. At its simplest, it is the study of how to represent functions as an infinite sum of sines and cosines. However, the evil mathematicians felt that Fourier analysis was too simple and decided to steal it from the hardworking physicists and expand on it. In addition, they decided to complicate it by introducing Harmonic analysis. So, what is the difference between Harmonic and Fourier analysis? We say that Harmonic analysis is the process of representing functions on a locally compact group G as a sum of the characters of the group, and Fourier analysis restricts this process to abelian groups.

In this talk I will introduce the Fourier series, the Fourier transform and how it applies to abelian groups. What about non-abelian groups, you might ask? The answer is Harmonic analysis! Finally, I will explain the role of Fourier transform in quantum mechanics, specifically how the Fourier transform interchanges position and momentum space.

### Basic Elliptic Curves

Delivered by Akshay Tiwary on Friday November 18, 2016

Elliptic Curves lie in the intersection of Algebra, Geometry, Number Theory and Complex Analysis. While my talk won't require any experience with complex analysis or algebraic geometry, I hope to expose you to this active area of research. Although I could tell you that the meat of Wiles' proof of Fermat's Last Theorem involved proving a special case conjecture previously known as the Taniyama Shimura conjecture involving elliptic curves, or that the Birch and Swinnerton-Dyer conjecture is an open millenium prize problem that talks about the relation between the arithmetic behaviour of elliptic curves and the analytic behaviour of an L-function, or that elliptic curves over finite fields is so useful for cryptography that there was a memo that recommended elliptic curves for federal government use in 1999, I don't have to because you're a math student who will attend this talk without needing to be wowed with cool applications, right?

### Simplicial Homology

Delivered by Kai Rüsch on Friday November 11, 2016

How do we count how many holes a shape has? We can answer this question using homology groups, whose order's count the number of $n$-dimensional holes.

A summary of this talk is available here.

### Integration

Delivered by Gregory Patchell on Friday October 28, 2016

The first part of this talk will be concerned with a brief introduction to measure theory. We will answer questions such a: How do we measure sets? Can every set be measured?

The second part of this talk will be the construction of the Lebesgue integral along with basic properties of the Lebesgue integral, along with comparisons to Riemann integration as we go. From Math 148, we saw that pointwise convergence doesn’t play well with Riemann integration. We will see that with the powerful Monotone Convergence Theorem and the Dominated Convergence Theorem, pointwise convergence and Lebesgue integration are a match made in heaven.

### Outline

• Definitions and examples of σ-algebras, measures, and measurable functions

• Motivations for Lebesgue integrals

• Construction of the Lebesgue integral with comparison to construction of the Riemann integral

• Benefits and properties of Lebesgue integration and limitations of Riemann integration

• Limitations of Lebesgue Integration

• Probability Spaces

• Vitali Sets

### Ultrafilters

Delivered by Felix Bauckholt on Wednesday October 12, 2016

In this talk, I will try to explain what ultrafilters are. I will do this by presenting a few motivating examples, and also some examples that, even if they don't motivate anything, are just really cool.

Hyperreal numbers are used to motivate this talk.

## Talk Suggestions

### Differential Equations on Fractals

Possible reference materials for this topic include

• Robert Strichartz : Differential Equations on Fractals - A tutorial

### Dirichlet's Theorem on Primes in Arithmetic progression

Possible reference materials for this topic include

### Integrals and Graph Theory

Possible reference materials for this topic include

### Iterated function systems

Aesthetically appealing fractal art can be created through interesting mathematical processes, including contraction mappings on metric spaces. Several well known fractals can be formalized mathematically and also appreciated visually. A talk on this subject would explore both mathematical and artistic aspects.

Possible reference materials for this topic include